Relations and Functions

Let $A = \{1, 2, 3\}$. A relation $R$ is defined on set $A$ as shown in the mapping diagram below. Determine which property the relation $R$ satisfies.

Let $A = \{1, 2, 3\}$. Which of the following is the smallest reflexive relation on $A$?

If $R$ is an equivalence relation on set $A$, it partitions $A$ into disjoint sets called:

Let $T$ be the set of all triangles in a plane. Let $R$ be the relation on $T$ defined by $aRb$ if $a$ is congruent to $b$. Which property is NOT satisfied by $R$?

Which property is violated by the relation $R = \{(1,2), (2,1)\}$ on the set $A = \{1, 2, 3\}$?

The relation $R = \{(a,b) : a, b \in \mathbb{Z}, a|b\}$ (a divides b) is:

Let $R$ be a relation defined on the set of natural numbers $\mathbb{N}$ by $x + 2y = 8$. Is $R$ transitive?

Let $S$ be a set of points in the plane. A relation $R$ is defined as $P_1 R P_2$ if the distance between $P_1$ and $P_2$ is less than 5 units. Given points $A, B, C$ on a line as shown, why is $R$ NOT an equivalence relation?

Consider the set of all people $P$. A relation $R$ is defined such that $a R b$ if $a$ is at least as tall as $b$. The diagram compares heights of three individuals $H_1, H_2, H_3$. Which property does this relation satisfy?

A relation $R$ on the set of natural numbers $\mathbb{N}$ is defined such that $x R y$ if $x + y$ is even. The diagram illustrates parity for even ($E$) and odd ($O$) numbers. Which statement is true for $R$?

Look at the graph of $f(x) = e^x$. If the codomain is $(0, \infty)$, is the function $f: \mathbb{R} \to (0, \infty)$ onto?

The diagram shows a function $f: A \to B$. If $f$ is one-to-one and onto, what is this type of function called?

Which of the following describes the function $f(x) = |x|$ for $f: \mathbb{R} \to \mathbb{R}$?

For a function $f: \{1, 2, 3\} \to \{4, 5\}$ as shown, is it possible for the function to be one-to-one?

In the mapping shown, if we restrict the codomain to only the set of images, what does the function become?

Consider $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = 2^x$. Is the function onto $\mathbb{R}$?

The function $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = x^3 + 2$. Using the graph, determine if $f$ is bijective.

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $f(x) = x^4 + x^2 + 1$. A researcher is studying the range and uniqueness of the output. Based on the behavior shown in the diagram, characterize this function.

A mapping $f: A \rightarrow B$ is illustrated by the following flow. Here, set $A = \{1, 2, 3\}$ and $B = \{a, b, c, d\}$. Based on the arrows, classify the function.

A physics student observes the velocity of a particle $v(t) = t^3 + t$ where $t \in \mathbb{R}$ and $v \in \mathbb{R}$. To determine if a specific velocity corresponds to a unique time, the student must check if the function is injective. Is the function $v(t)$ one-to-one and onto?

Consider the function $f: [0, \infty) \to [0, \infty)$ defined by $f(x) = x^2$, as shown in the diagram. If $g(x)$ is the inverse of $f(x)$, what is the value of $g(9)$?

If $f(x) = x^2$ is restricted to the domain $x \ge 0$, find $f^{-1}(9)$.

The point $(a, b)$ lies on the graph of a one-to-one function $f$. Which point must lie on the graph of $f^{-1}$?

If $f(x) = \frac{1}{x}$, what is $f^{-1}(x)$?

The function $f(x) = \frac{x+1}{x-2}$ is defined for $x \neq 2$. Find the value of $k$ such that $f^{-1}(x) = \frac{2x+1}{x-k}$.

Find the inverse of the function $f(x) = \frac{e^x}{1+e^x}$.

The diagram shows a rectangle with vertices at the origin and on the line $y = -x + 4$. If we define a function $f$ mapping the x-coordinate to the y-coordinate on this line, what is $f^{-1}(x)$?

A population growth model is defined by $P(t) = 5 + 2^{t/3}$. Determine the time $t$ in terms of the population $P$.

A flow control valve's opening degree $y$ is related to the pressure $x$ by $y = \frac{2x}{1+x}$ for $x \ge 0$. Find the pressure $x$ as a function of the opening degree $y$.

Consider the function $f: \mathbb{R} \to (0, \infty)$ defined by $f(x) = 10^{2x-4}$. Find the inverse function $f^{-1}(x)$ used in a logarithmic scaling tool.